An object S is terminal in a category if for each object X in the category there is exactly one map from X to S. Only the free category 𝑭(G_b) on the graph (b) and the free category 𝑭(G_c) on the graph (c) have objects that meet this criterion.

(a) 𝑭(G_a) has infinitely many maps from the object to itself (each map corresponding to the path formed by going around the arrow/loop a different number of times).

(b) The only object in 𝑭(G_b) is terminal, since there is exactly one map from that object to itself, namely the identity map.

(c) In 𝑭(G_c), the second dot from the right corresponds to a terminal object, since there is exactly one map (path) to that object from every object in 𝑭(G_c). We label the dots from left to right, with x_1 being the upper left dot and x_2 the lower left dot (so x_4 is terminal).

(d) 𝑭(G_d) is similar to 𝑭(G_a) in that each object of 𝑭(G_d) has infinitely many maps from itself to itself and from the other object to itself (each map corresponding to the path formed by going a different number of times around the closed loop of arrows between the two objects).

(e) In 𝑭(G_e), the left-hand object has no map from the right-hand object, while the right-hand object has two maps from the left-hand object. We label the dots from left to right.